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Many types of equations can be solved using 'normal' techniques.
- Linear equations can be solved by rearrangement
- Quadratic equations can be solved using the quadratic formula
- More complex equations can be solved by factorising by inspection, or by using the Factor Theorem.
Let's look at an example to show this:
Solve: x3 = x2 + 7x + 20
Firstly rewrite the equation as f(x) = 0
Therefore: x3 - x2 - 7x - 20 = 0
This has no obvious factors, so use the factor theorem. If f(a) = 0 then (x − a) is a factor. We then try values for a.
f(4) = 64 - 16 - 28 - 20 = 0. Therefore (x - 4) is a factor.
To find the other solutions we know that x3 - x2 - 7x - 20 can now be written as...
x3 - x2 - 7x - 20
= (x − 4)(x2 + bx + c)
= x3 + (b − 4)x2 + (c − 4b)x − 4c
Matching the expressions gives us:
b − 4 = -1, so b = 3, and
c − 4b = -7, so c = 5.
x3 - x2 - 7x - 20 = (x − 4)(x2 + 3x + 5)
The quadratic in this case does not factorise (b2 - 4ac < 0). So there is only one solution to the equation: x = 4.
When this technique does not work we need a new method. Numerical Methods are methods that can be used in these cases.
Numerical Methods are systems, or algorithms, for solving equations that cannot be solved using normal techniques. There are a number of different types of numerical methods available. The ones you need for your exams are listed below and are shown in more detail in the next Learn-It:
- Change of the Sign Methods
- Newton Raphson